MCQ
The product of three consecutive positive integers is divisible by :
- ✓$6$
- B$4$
- C$5$
- D$10$
✓
Answer
Correct option: A.
$6$
Let $n$ be a positive integer,
then three consecutive positive integers are $(n + 1)(n + 2)(n + 3) $
$= n(n + 1)(n + 2) +3 (n + 1)(n + 2)$
Here, the first term is divisible by $6$ and the second term is also divisible by $6$
Because it contains a factor $3$ and one of the two consecutive integers $(n + 1)$ or $(n + 2$) is even and thus is divisible by $2$.
$\therefore,$ the sum of multiple of $6$ is also a multiple of $6$.
then three consecutive positive integers are $(n + 1)(n + 2)(n + 3) $
$= n(n + 1)(n + 2) +3 (n + 1)(n + 2)$
Here, the first term is divisible by $6$ and the second term is also divisible by $6$
Because it contains a factor $3$ and one of the two consecutive integers $(n + 1)$ or $(n + 2$) is even and thus is divisible by $2$.
$\therefore,$ the sum of multiple of $6$ is also a multiple of $6$.
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